How to show (only) portions of Wireframe modifier? (Version 2.8)
How about, instead of using the wireframe modifier:


Deleting all but one of the radial edges of your cone, cut to the right length
Alt C convert it to a curve
In the curve's data panel set its full bevel to your liking
Make an empty at center of the cone (its original origin) rotated in Z by 360/32
Make an array of your curve, count 32, Object Offset, the empty as target.


Now you can keyframe the visibility of the original curve (if you want a complete disappearance). For the rest, you can keyframe the Count of the Array modifier.